College Algebra Exam Review 372

College Algebra Exam Review 372 - 382 8. MODULES If S Â M...

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Unformatted text preview: 382 8. MODULES If S Â M is any subset, we define \ annihilator of S to be ann.S / D the fr 2 R W rx D 0 for all x 2 S g D ann.x/: Note that ann.S / is an x 2S ideal, and ann.S / D ann.RS /, the annihilator of the submodule generated by S . See Exercise 8.5.1 Consider a torsion module M over R. If S is a finite subset of M then ann.S/ D ann.RS / is a nonzero ideal of R; in fact, if S D fx1 ; : : : ; xn g and for each i , ri is a nonzero element of R such that ri xi D 0, then Q i ri is a nonzero element of ann.S /. If M is a finitely generated torsion module, it follows that ann.M / is a nonzero ideal of R. For the remainder of this section, R again denotes a principal idea domain and M denotes a (nonzero) finitely generated module over R. For x 2 Mtor , any generator of the ideal ann.x/ is called a period of x . If a 2 R is a period of x 2 M , then Rx Š R=ann.x/ D R=.a/. According to Lemma 8.4.5, any submodule of M is finitely generated. If A is a torsion submodule of M , any generator of ann.A/ is a called a period of A. The period of an element x , or of a submodule A, is not unique, but any two periods of x (or of A) are associates. Lemma 8.5.5. Let M be a finitely generated module over a principal ideal domain R. (a) If M D A ˚ B , where A is a torsion submodule, and B is free, then A D Mtor . (b) M has a direct sum decomposition M D Mtor ˚ B , where B is free. The rank of B in any such decomposition is uniquely determined. (c) M is a free module if, and only if, M is torsion free. Proof. We leave part (a) as an exercise. See Exercise 8.5.3. According to the existence part of Theorem 8.5.2, M has a direct sum decomposition M D A ˚ B , where A is a torsion submodule, and B is free. By part (a), A D Mtor . Consequently, B Š M=Mtor , so the rank of B is determined. This proves part (b). For part (c), note that any free module is torsion free. On the other hand, if M is torsion free, then by the decomposition of part (b), M is free. I ...
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